The following is an incomplete table for the relation \(y = 2x^{2} - 5x + 1\) x -3 -2 -1 0 1 2 3 4 5 y 8 1 -1 26 (a) Copy and complete the table. (b) Using ...
Question 1 Report
The following is an incomplete table for the relation \(y = 2x^{2} - 5x + 1\)
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
| y | 8 | 1 | -1 | 26 |
(a) Copy and complete the table.
(b) Using a scale of 2cm to 1 unit on the x- axis and 2cm to 10 units on the y- axis, draw the graph of the relation \(y = 2x^{2} - 5x + 1\) for \(-3 \leq x \leq 5\).
(c) Using the same scale and axes, draw the graph of \(y = x + 6\).
(d) Estimate from your graphs, correct to one decimal place : (i) the least value of y and the value of x for which it occurs ; (ii) the solution of the equation \(2x^{2} - 5x + 1 = x + 6\).
Answer Details
(a) Completing the table for \(y = 2x^{2} - 5x + 1\)
Substitute each x-value, for example \(x=-3:\; 2(9)-5(-3)+1 = 18+15+1 = 34\).
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|---|---|---|
| y | 34 | 19 | 8 | 1 | -2 | -1 | 4 | 13 | 26 |
(b) The graph
Plot the nine points with 2 cm to 1 unit on the x-axis and 2 cm to 10 units on the y-axis, and join them with a smooth U-shaped parabola.
(c) The line \(y = x + 6\)
Use two points: at \(x=-3,\;y=3\) and at \(x=5,\;y=11\). Draw the straight line through \((-3,3)\) and \((5,11)\).
(d)(i) Least value of y
The lowest point of the parabola occurs at the vertex, \(x = \dfrac{5}{2\times 2} = 1.25\). Then
\(y = 2(1.25)^{2} - 5(1.25) + 1 = 3.125 - 6.25 + 1 = -2.125\).
From the graph, the least value of y is \(\approx -2.1\), occurring at \(x \approx 1.3\).
(d)(ii) Solution of \(2x^{2} - 5x + 1 = x + 6\)
The solutions are the x-coordinates where the curve meets the line. Algebraically:
\(2x^{2} - 5x + 1 = x + 6 \Rightarrow 2x^{2} - 6x - 5 = 0\).
\(x = \dfrac{6 \pm \sqrt{36+40}}{4} = \dfrac{6 \pm \sqrt{76}}{4}\), giving \(x \approx 3.7\) and \(x \approx -0.7\).
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